There has been considerable literature over the past ten years pertaining to the properties and possible devices based on photonic crystal structures. In particular, there has been some recognition in the prior art that certain magnetic photonic crystal structures have the potential to create spatial frequency asymmetry for light propagation in different directions through the crystal. Such structures could provide the basis for a whole new class of optical devices. For example, such a crystal structure might be used as an optical memory device where the group velocity property of light propagating in one direction through the crystal is reduced to a near-zero speed upon the application of a magnetic field. Such a crystal structure might also be used for high speed modulation or demodulation of an optical signal, or as an optical routing or switching device.
However, fabrication of the periodic dielectric patterns required for such photonic crystals has proved problematical, especially for 3-D structures. Such periodic dielectric patterns in the crystal structure produces “gaps” in frequency where propagation is forbidden. This effect is readily seen through the use of a frequency vs. propagation constant diagram. (See J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals—Molding The Flow of Light,” Princeton University Press, 1995). The specifics of such diagrams depend on the particular geometric pattern of areas having different dielectric constants as well as the relative difference between the constants. In the examples in this application, the required periodic dielectric pattern is achieved by a periodic square array of holes in a dielectric material. By the incorporation of appropriate disruptions of the periodicity, one can create “localized” states within the gaps. These localized states can lead to waveguides, or resonator structures.
In general the frequency vs propagation constant diagram is symmetric such that ω(k)=ω(−k). This conclusion can be drawn by following the approach given by Joannopoulas (Photonic Crystals, page 36) to establish the time invariance of the energy bands. Using his notation for the Maxwell operator
                    Θ        =                              ∇                          x              ⁡                              (                                                      1                                          g                      ⁡                                              (                        r                        )                                                                              ∇                                )                                              ⁢          x                                    1        )            he writes the operator equationΘHk=(ω/c)2Hk  2)Taking the complex conjugate of equation 2 and noting that Θ=Θ*, one hasΘHk*=(ω/c)2Hk  3)This shows that Hk* is also an eigenvector of Θ with the same eigenvalue as Hk. From the Bloch representation of Hk Hk=exp(k·r)u(r)  4)One sees that Hk* corresponds to the wave traveling backward with propagation vector −k. The conclusion is that since both Hk, and Hk* have the same eigenvalues, then one must conclude thatω(k)=ω(−k)  5)The explicit assumption in this derivation is that the dielectric tensor “∈” is real. In the above case it is of the simplest form of a scalar. The more general case occurs when there is a magnetic field present, either external, or internal. As an example, consider the situation of an otherwise isotropic medium in a static magnetic field. The dielectric tensor is now of the form (see A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation, John Wiley & Sons, Hoboken, N.J., 1983)D=∈E+iγBxE  6)The γ for an isotropic material is a scalar, but in the more general case it is a tensor whose form depends on the symmetry of the material. One can arrive at a similar expression for D from a free-energy expansion in E and H.φ=∈ijEiEj+μijHiHj+γijkBiEjEk+ . . .  7)Here D=∂φ/∂E, and γ is complex. For the isotropic case, one applies the appropriate symmetry operations to obtain the non-zero elements of γ.
If one expands equation 6 for the case of a static magnetic field in the z-direction one has the following result
                              D          i                =                              (                                                                                D                    x                                                                                                                    D                    y                                                                                                                    D                    z                                                                        )                    =                                                    (                                                                            ɛ                                                                                                                -                          i                                                ⁢                                                                                                  ⁢                        γ                        ⁢                                                                                                  ⁢                                                  B                          z                                                                                                            0                                                                                                                          i                        ⁢                                                                                                  ⁢                        γ                        ⁢                                                                                                  ⁢                                                  B                          z                                                                                                            ɛ                                                              0                                                                                                  0                                                              0                                                              ɛ                                                                      )                            ⁢                              (                                                                                                    E                        x                                                                                                                                                E                        y                                                                                                                                                E                        z                                                                                            )                                      =                                          ɛ                ij                            ⁢                              E                j                                                                        8        )            
In the case of no external magnetic field, the dielectric tensor is symmetric, but the general condition that is required in the case of no absorption is that it be Hermitian.∈ij=└∈ji*┘  9)
The expression for the dielectric tensor in equation 8 is now the one we will use in the equation 1 for the Maxwell operator. One finds now that although Θ is still Hermitian, (conjugate transpose) nonetheless Θ≠Θ*. We see that the representation of the time reversal operation with conjugation is the condition that constitutes time reversal in its simplest form.
To consider time reversal in the more general case, and in particular the consequence it may have on spectral asymmetry, one must consider its properties in more detail. It is clear that for the anti-symmetric ω(k)=−ω(−k) condition to hold then both time reversal and spatial inversion must not be elements of the symmetry group of the crystal. The situation has been considered for the case of the degeneracy of energy bands in magnetically ordered crystals. Time reversal, in addition to t→−t, has the effect of reversing the direction of the spin. For crystals exhibiting magnetic ordering (either internal or through an external magnetic field), the symmetry classification has to be expanded to take this condition into account. The fact that the time invariance operator is non-unitary does not allow for a simple representation, as exists for the spatial symmetry operations.
The invention is based on the applicants' recognition that there are 2-D photonic crystal structures with anti-symmetric mode behavior, which largely solves the fabrication problem referred to earlier. A. Figotin and I. Vitebsky, Non-reciprocal Magnetic Photonic Crystals, Phys. Rev. E., Vol. 63, 066609 (2001) show some computed examples of spectral asymmetry for the simple 1-D structure. Here he uses alternate layers of magnetic and non-magnetic materials are used to create a dielectric reflector. To have spectral asymmetry in a 1-D structure requires additional anisotropy in the non-magnetic layer. As discussed hereinafter, this condition may be obviated in higher dimensional structures by the inclusion of an appropriate magnetic medium or external field. It should be noted that in this case the time invariance breaking magnetic field is an internal one as provided by the ordered magnetic structure. In the derivation given above, the use of an external field is assumed. This case is more general in that it does not require a ferromagnetic medium. However, from a practical view the degree of spectral asymmetry will depend on the magnitude of the field which in the case of a ferromagnetic material, the internal field, “B” will be quite large.